Séminaire Lotharingien de Combinatoire, 78B.45 (2017), 12 pp.

Thomas Gobet

On Cycle Decompositions in Coxeter Groups

Abstract. The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, allowing a generalization of it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasi-Coxeter element). We show that such an element admits an analogue of the cycle decomposition. Elements which are not in this family still admit a generalized cycle decomposition, but it is not unique in general.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

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